Comprehensive Review of Neural Differential Equations for Time Series Analysis

Overview

Time series modeling and analysis have become critical in various domains. Conventional methods such as RNNs and Transformers, while effective for discrete-time and regularly sampled data, face significant challenges in capturing the continuous dynamics and irregular sampling patterns inherent in real-world scenarios. Neural Differential Equations (NDEs) represent a paradigm shift by combining the flexibility of neural networks with the mathematical rigor of differential equations. This paper presents a comprehensive review of NDE-based methods for time series analysis, including neural ordinary differential equations, neural controlled differential equations, and neural stochastic differential equations. We provide a detailed discussion of their mathematical formulations, numerical methods, and applications, highlighting their ability to model continuous-time dynamics. Furthermore, we address key challenges and future research directions. This survey serves as a foundation for researchers and practitioners seeking to leverage NDEs for advanced time series analysis.

This survey provides a comprehensive review of NDE-based methods for time series analysis, covering the three core families:

Neural Ordinary Differential Equations (NODEs)
Neural Controlled Differential Equations (NCDEs)
Neural Stochastic Differential Equations (NSDEs)

We offer a detailed discussion of their mathematical formulations, numerical methods, and key applications, highlighting their unique ability to model continuous-time dynamics. This survey serves as a foundational guide for researchers and practitioners aiming to leverage NDEs for advanced time series analysis.

Key Contributions of This Survey

This paper provides a structured synthesis of the NDE field for time series analysis, with a focus on:

Foundational Models: A detailed breakdown of the formulations and extensions for NODEs, NCDEs, and NSDEs.
Theoretical Underpinnings: An analysis of universal approximation properties, conditions for the existence and uniqueness of solutions, and the stability of different NDE families.
Practical Implementation: A guide to practical considerations, including optimization with the adjoint method, regularization techniques, and the selection of numerical solvers.
Empirical Comparison: A consolidated summary of benchmark results across a wide range of tasks, including interpolation, forecasting, and classification.